Nonorthogonal Localized Molecular Orbitals Research Articles

Regularized Localized Molecular Orbitals in a Divide-and-Conquer Approach for Linear Scaling Calculations.

Non-orthogonal localized molecular orbitals (NOLMOs) have been employed as building blocks for the divide-and-conquer (DC) linear scaling method. The NOLMOs are calculated from subsystems and used for constructing the density matrix (DM) of the entire system, instead of the subsystem DM in the original DC approach. Also, unlike the original DC method, the inverse electronic temperature parameter β is not needed anymore. Furthermore, a new regularized localization approach for NOLMOs has been developed, in which the localization cost function is a sum of the spatial spread function, as in the Boys method, and the kinetic energy, as a regularization measure to limit the oscillation of the NOLMOs. The optimal weight of the kinetic energy can be determined by optimization with analytical gradients. The resulting regularized NOLMOs have enhanced smoothness and better transferability because of reduced kinetic energies. Compared with the original DC, while NOLMO-DC has a similar computational linear scaling cost, the accuracy of NOLMO-DC is better by several orders of magnitude for large conjugated systems and by about 1 order of magnitude for other systems. The NOLMO-DC method is thus a promising development of the DC approach for linear scaling calculations.

Variable-Metric Localization of Occupied and Virtual Orbitals.

The key idea of the variable-metric approach to orbital localization is to allow nonorthogonality between orbitals while, at the same time, preventing them from becoming linearly dependent. The variable-metric localization has been shown to improve the locality of occupied nonorthogonal orbitals relative to their orthogonal counterparts. In this work, numerous localization algorithms are designed and tested to exploit the conceptual simplicity of the variable-metric approach with the goal of creating a straightforward and reliable localization procedure for virtual orbitals. The implemented algorithms include the steepest descent, conjugate gradient (CG), limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), and hybrid procedures as well as trust-region (TR) methods based on the CG and Cauchy-point subproblem solvers. Comparative analysis shows that the CG-based TR algorithm is the best overall method to obtain nonorthogonal localized molecular orbitals (NLMOs), occupied or virtual. The L-BFGS and CG algorithms can also be used to obtain NLMOs reliably but often at higher computational cost. Extensive tests demonstrate that the implemented methods allow us to obtain well-localized Boys-Foster (i.e., maximally localized Wannier functions) and Pipek-Mezey, orthogonal and nonorthogonal, and occupied and virtual orbitals for a variety of gas-phase molecules and periodic materials. The tests also show that virtual NLMOs, which have not been described before, are, on average, 13% (Boys-Foster) and 18% (Pipek-Mezey) more localized than their orthogonal counterparts.

Time-Dependent Coupled Perturbed Hartree-Fock and Density-Functional-Theory Approach for Calculating Frequency-Dependent (Hyper)Polarizabilities with Nonorthogonal Localized Molecular Orbitals.

The time-dependent coupled perturbed Hartree-Fock/density-functional-theory (TDHF/TDDFT) approach has been reformulated based on nonorthogonal localized molecular orbitals (NOLMOs). Based on the NOLMO Fock equation, we have derived the corresponding NOLMO-TDHF/TDDFT equations up to the third order, and the formula for the frequency-dependent (hyper)polarizabilities has been given. Our approach has been applied to calculate both static and dynamic (hyper)polarizabilities of molecules varying from small molecules to large molecules. The NOLMO-TDHF/TDDFT approach can reproduce the reference canonical molecular orbital (CMO) results for all of our testing calculations. With the help of ongoing development of optimized local virtual molecular orbitals, the NOLMO-TDHF/TDDFT approach would be a very efficient method for large system calculations and tp achieve linear scaling.

Coupled-Perturbed SCF Approach for Calculating Static Polarizabilities and Hyperpolarizabilities with Nonorthogonal Localized Molecular Orbitals.

Coupled-perturbed self-consistent-field (CPSCF) approach has been broadly used for polarizabilities and hyperpolarizabilities computation. To extend this application to large systems, we have reformulated the CPSCF equations with nonorthogonal localized molecular orbitals (NOLMOs). NOLMOs are the most localized representation of electronic degrees of freedom. Methods based on NOLMOs are potentially ideal for investigating large systems. In atomic orbital representation, with a static external electric field added, the wave function and SCF operator of unperturbed NOLMO-SCF wave function/orbitals are expanded to different orders of perturbations. We have derived the corresponding equations up to the third order, which are significantly different from those of a conventional CPSCF method because of the release of the orthogonal restrictions on MOs. The solution to these equations has been implemented. Several chemical systems are used to verify our method. This work represents the first step toward efficient calculations of molecular response and excitation properties with NOLMOs.

Efficient Linear-Scaling Density Functional Theory for Molecular Systems.

Despite recent progress in linear scaling (LS) density function theory (DFT), the computational cost of the existing LS methods remains too high for a widespread adoption at present. In this work, we exploit nonorthogonal localized molecular orbitals to develop a series of LS methods for molecular systems with a low computational overhead. High efficiency of the proposed methods is achieved with a new robust two-stage variational procedure or by replacing the optimization altogether with an accurate nonself-consistent approach. We demonstrate that, even for challenging condensed-phase systems, the implemented LS methods are capable of extending the range of accurate DFT simulations to molecular systems that are an order of magnitude larger than those previously treated.

Effective preconditioning for ab initio ground state energy minimization with non-orthogonal localized molecular orbitals

The non-orthogonal localized molecular orbital (NOLMO) is the most localized representation of electronic degrees of freedom. As such, NOLMOs are thus potentially the most efficient for linear-scaling calculations of electronic structures for large systems. However, direct ab initio calculations with NOLMO have not been fully implemented and widely used, partly because of the slow convergence issue in the optimization of NOLMO. Towards realizing the potential of NOLMO for large systems, we applied an energy minimum variational principle for carrying out ab initio self-consistent-field (SCF) calculations with NOLMOs. We developed an effective preconditioning approach using the diagonal part of the second order derivatives and show that the convergence of the energy optimization is significantly improved. The speed of convergence of the energy and density are comparable with that of the conventional SCF approach, thus paving the way for the optimization of NOLMO in linear scaling calculations for large systems.

ChemInform Abstract: Linear‐Scaling Quantum Mechanical Methods for Excited States

AbstractReview: 70 refs.

Linear-scaling quantum mechanical methods for excited states

The poor scaling of many existing quantum mechanical methods with respect to the system size hinders their applications to large systems. In this tutorial review, we focus on latest research on linear-scaling or O(N) quantum mechanical methods for excited states. Based on the locality of quantum mechanical systems, O(N) quantum mechanical methods for excited states are comprised of two categories, the time-domain and frequency-domain methods. The former solves the dynamics of the electronic systems in real time while the latter involves direct evaluation of electronic response in the frequency-domain. The localized density matrix (LDM) method is the first and most mature linear-scaling quantum mechanical method for excited states. It has been implemented in time- and frequency-domains. The O(N) time-domain methods also include the approach that solves the time-dependent Kohn-Sham (TDKS) equation using the non-orthogonal localized molecular orbitals (NOLMOs). Besides the frequency-domain LDM method, other O(N) frequency-domain methods have been proposed and implemented at the first-principles level. Except one-dimensional or quasi-one-dimensional systems, the O(N) frequency-domain methods are often not applicable to resonant responses because of the convergence problem. For linear response, the most efficient O(N) first-principles method is found to be the LDM method with Chebyshev expansion for time integration. For off-resonant response (including nonlinear properties) at a specific frequency, the frequency-domain methods with iterative solvers are quite efficient and thus practical. For nonlinear response, both on-resonance and off-resonance, the time-domain methods can be used, however, as the time-domain first-principles methods are quite expensive, time-domain O(N) semi-empirical methods are often the practical choice. Compared to the O(N) frequency-domain methods, the O(N) time-domain methods for excited states are much more mature and numerically stable, and have been applied widely to investigate the dynamics of complex molecular systems.

Efficient Construction of Nonorthogonal Localized Molecular Orbitals in Large Systems†

Localized molecular orbitals (LMOs) are much more compact representations of electronic degrees of freedom than canonical molecular orbitals (CMOs). The most compact representation is provided by nonorthogonal localized molecular orbitals (NOLMOs), which are linearly independent but are not orthogonal. Both LMOs and NOLMOs are thus useful for linear-scaling calculations of electronic structures for large systems. Recently, NOLMOs have been successfully applied to linear-scaling calculations with density functional theory (DFT) and to reformulating time-dependent density functional theory (TDDFT) for calculations of excited states and spectroscopy. However, a challenge remains as NOLMO construction from CMOs is still inefficient for large systems. In this work, we develop an efficient method to accelerate the NOLMO construction by using predefined centroids of the NOLMO and thereby removing the nonlinear equality constraints in the original method ( J. Chem. Phys. 2004 , 120 , 9458 and J. Chem. Phys. 2000 , 112 , 4 ). Thus, NOLMO construction becomes an unconstrained optimization. Its efficiency is demonstrated for the selected saturated and conjugated molecules. Our method for fast NOLMO construction should lead to efficient DFT and NOLMO-TDDFT applications to large systems.

Reformulating time-dependent density functional theory with non-orthogonal localized molecular orbitals

Time-dependent density functional theory (TDDFT) has broad application in the study of electronic response, excitation and transport. To extend such application to large and complex systems, we develop a reformulation of TDDFT equations in terms of non-orthogonal localized molecular orbitals (NOLMOs). NOLMO is the most localized representation of electronic degrees of freedom and has been used in ground state calculations. In atomic orbital (AO) representation, the sparsity of NOLMO is transferred to the coefficient matrix of molecular orbitals (MOs). Its novel use in TDDFT here leads to a very simple form of time propagation equations which can be solved with linear-scaling effort. We have tested the method for several long-chain saturated and conjugated molecular systems within the self-consistent charge density-functional tight-binding method (SCC-DFTB) and demonstrated its accuracy. This opens up pathways for TDDFT applications to large bio- and nano-systems.

Linear-scaling quantum calculations using non-orthogonal localized molecular orbitals

An absolute energy minimum variational principle is used for carrying out linear-scalingcalculations with non-orthogonal localized orbitals. Comparing with results based onorthogonal localized molecular orbitals, the method is shown to give significantly moreaccurate results when the localized molecular orbitals are allowed to be non-orthogonal.This is made possible by introducing a second minimization for approximatingthe inverse overlap matrix. We also show how an exact line search may be usedefficiently with the conjugate gradient method for minimizing the energy functional.

LocalSCF method for semiempirical quantum-chemical calculation of ultralarge biomolecules.

A linear-scaling semiempirical method, LocalSCF, has been proposed for the quantum-chemical calculations of ultralarge molecular systems by treating the large-scale molecular task as a variational problem. The method resolves the self-consistent field task through the finite atomic expansion of weakly nonorthogonal localized molecular orbitals. The inverse overlap matrix arising from the nonorthogonality of the localized orbitals is approximated by preserving the first-order perturbation term and applying the second-order correction by means of a penalty function. This allows for the separation of the orbital expansion procedure from the self-consistent field optimization of linear coefficients, thereby maintaining the localized molecular orbital size unchanged during the refinement of linear coefficients. Orbital normalization is preserved analytically by the variation of virtual degrees of freedom, which are orthogonal to the initial orbitals. Optimization of linear coefficients of localized orbitals is performed by a gradient procedure. The computer program running on a commodity personal computer was applied to the GroEL-GroES chaperonin complex containing 119,273 atoms.

An efficient method for constructing nonorthogonal localized molecular orbitals.

A new method for constructing nonorthogonal localized molecular orbitals (NOLMOs) is presented. The set of highly localized NOLMOs is obtained by minimization of the spread functional starting from an initial set of canonical orthogonal molecular orbitals. To enhance the stability and efficiency, the centroids of the NOLMOs are constrained to be those of the corresponding orthogonal localized molecular orbitals (OLMOs), which are obtained with the Boys criterion in advance. In particular, these centroid constraints make the optimization for each NOLMO independent of the others, which is an attractive feature for application to large systems. The minimization with the constraints incorporated through the multiplier-penalty function method is stable and efficient in convergence. While exhibiting the classical bonding pattern in chemistry and sharing a spatial distribution similar to that of the corresponding OLMOs, the obtained NOLMOs are more compact than the corresponding OLMOs with about 10%-28% reduction in the value of the spread functional and devoid of the troublesome "orthogonalization tails."

Study on the construction of satisfactory nonorthogonal localized molecular orbitals

Comparing to orthogonal localized molecular orbitals (OLMO), the nonorthogonal localized molecular orbitals (NOLMO) exhibit bonding pictures more accordant with those in the traditional chemistry. They are more contracted, so that they have a better transferability and better performances for the calculation of election correlation energies and for the linear scaling algorithms of large systems. The satisfactory NOLMOs should be as contracted as possible while their shapes and spatial distribution keep in accordance with the traditional chemical bonding picture. It is found that the spread of NOLMOs is a monotonic decreasing function of their orthogonality, and it may reduce to any extent as the orthogonality descends. However, when the orthogonality descends to some point, the shapes and spatial distribution of the NOLMOs deviate drastically from the traditional chemical bonding picture, and finally the NOLMOs tend to linear dependence. Without the requirement of orthogonalization, some other constraints have to be imposed for constructing satisfactory NOLMOs by minimizing their spread functional. It is shown that satisfactory results can be generated by coupling the minimization of orbital spread functionals with the maximization of the distances between orbital centroids.

Intermolecular exchange-induction and charge transfer: Derivation of approximate formulas using nonorthogonal localized molecular orbitals

Previous work on the efficient evaluation of the second-order exchange repulsion energy [Mol. Phys. 89, 1313 (1996)] is extended to exchange induction and charge transfer energies. The approximations outlined in the previous work are shown to be sufficient to simplify all new terms in the exchange induction and charge transfer energy terms. Both variational and couple perturbed Hatree–Fock approaches are discussed as is the use of these new energy terms in conjunction with hybrid “quantum mechanical/molecular mechanics” methods [specifically the effective fragment potential method, J. Chem. Phys. 105, 1968 (1996)].

Nonorthogonal localized molecular orbitals in electronic structure theory

The concept of nonorthogonal localized molecular orbital (NOLMO) is investigated in this paper. Given a set of the commonly used canonical molecular orbitals, a direct minimization algorithm is proposed to obtain both the orthogonal localized molecular orbitals (OLMO) and NOLMO by using the Boys criterion and conjugate gradient minimization. To avoid the multiple-minimum problem, the absolute energy minimization principle of Yang is employed to obtain initial guesses. Contrary to the early conclusion drawn by Lipscomb and co-workers who claimed that OLMOs and the corresponding NOLMOs are more or less the same, we found that NOLMOs are about 10%–30% more localized than OLMOs. More importantly, the so-called “delocalization tail” that plagues OLMOs is not present in NOLMOs, showing that NOLMOs are more compact and less oscillatory and capable of providing greater transferability in describing the electronic structure of molecules. We also found that main lobes of NOLMOs are slightly larger in size than those of OLMOs because of the normalization requirement. These features establish NOLMOs to be valuable as building blocks in electronic structure theory and for the understanding of chemical bonding. They show the promise for the utilization of NOLMOs—the most localized possible—in the linear scaling approaches of the electronic structure theory for molecules and solids.

Increased-valence structures and hypervalent molecules

Hypervalent molecules may involve the use of increased-valence structures to provide valence bond descriptions of their electronic structure. For electron-rich molecules with four electrons distributed among three overlapping (nuclear-centered) atomic orbitals, the increased-valence structures are Y(BOND)A·B and Y·A——B. Each structure involves a fractional electron-pair bond and a one-electron bond. It is deduced that the Armstrong-Perkins-Stewart valence of the A atom is able to exceed unity in each of these structures when the three bonding electrons occupy nonorthogonal localized molecular orbitals. It is also shown that increased valence for the A atom does not occur when the four electrons occupy localized molecular orbitals to give the valence-bond structure Y—A—B with three overlapping atomic orbitals, and the same number of orbital variational parameters as occurs in the wave functions for either of the increased-valence structures. The results of ab initio valence bond calculations with minimal basis sets are reported for H−3l, CH−5, HF−2, F−3, CIF3, and FF3, and the resulting wave functions for resonance between six canonical Lewis structures are related to those for resonance between the two increased-valence structures. The use of the latter structures to indicate how electronic reorganization proceeds via one-electron delocalizations for SN2 reactions is redescribed, and an elementary argument is presented to deduce that this class of reactions cannot involve the delocalization of a pair of electrons in concert from the nucleophile. Increased-valence wave functions are used to deduce an expression for the avoided crossing for the transition state of the identity SN2 reaction. © 1996 John Wiley & Sons, Inc.

Non-orthogonal localized orbitals and orthogonal atomic hybrids derived from Mulliken's population analysis

The construction of non-orthogonal localized molecular orbitals (Wannier functions of extended systems) with maximal degree of localization according to the Magnasco-Perico localization criterion derived from Mulliken's population analysis is discussed and several forms of the equations giving these orbitals are presented. A partial orthogonality property of these orbitals permits an easy determination of the effective orthogonal atomic orbitals characterizing the actual state of the atom in the molecule; this helps to find connections between large-scale ab initio SCF calculations and traditional atomic orbital concepts. Calculations indicate full agreement with the traditional AO picture: for all ‘ordinary’ compounds one gets as many molecular orbitals appreciably localized on the given atom as are contained in the classical ‘minimal basis’ for that atom.

Decomposition of electron density matrix and total energy of supermolecule in the method of non-orthogonal localized molecular orbitals

The decomposition of the electron density matrix and the total energy of a supermolecule in the one-configuration approximation of the non-orthogonal localized molecular orbitals (NLMO) method based on the Frobenius block inversion formulae of the NLMO overlap matrix for two and many interacting subsystems was considered in terms of reduced one-particle density matrices of the subsystems. Formulae were derived which can be used as a theoretical basis for a qualitative and quantitative analysis of the coulombic, induction and exchange components of the intermolecular and intramolecular interaction energies and for the development of models of such interactions. This approach is related to the decomposition method developed by Morokuma and Umeyama, and to the method of non-orthogonal group functions of McWeeny and Dacre.

Theoretical investigations of the nature of intramolecular interactions

The use of non-orthogonal localized molecular orbitals (NOLMOs) for the construction of the one-electron density of closed shell molecules is investigated. Special attention is focused on the trans...